buildings
Article
Analysis of Structural Layouts of Geodesic Dome Structures
with Bar Filler Considering Air Transportation
Andrey Kolpakov 1, * , Oleg Dolgov 1 , Vladislav Korolskiy 1 , Semen Popov 1 , Vyacheslav Anchutin 1
and Vadim Zykov 2
1
2
*
Citation: Kolpakov, A.; Dolgov, O.;
Korolskiy, V.; Popov, S.; Anchutin, V.;
Zykov, V. Analysis of Structural
Layouts of Geodesic Dome Structures
with Bar Filler Considering Air
Moscow Aviation Institute, National Research University, Volokolamskoe Shosse, 4, 125993 Moscow,
Moscow Region, Russia; dolgov@mai.ru (O.D.); korolskijvv@mai.ru (V.K.); semen-popov-2001@mail.ru (S.P.);
vyacheslav.anchutin@mail.ru (V.A.)
All-Russian Research Institute for Fire Protection of the Ministry of the Russian Federation for Civil Defence,
Emergencies and Elimination of Consequences of Natural Disasters, mkr. VNIIPO, 12,
143903 Balashikha, Moscow Region, Russia; otdel-15@vniipo.ru
Correspondence: a.kolpakov@mai.ru
Abstract: The results are presented from a study of three-layer geodesic dome structures with bar
fillers under their own weight. An algorithm was developed for selecting the type of structural
layout used and the reference parameters chosen in terms of the technological, strength, and weight
characteristics. The results of this analysis aim to make it easier for designers to determine the
optimal reference parameters in the initial stage of the designing of geodetic hemispherical dome
structures, the construction of which is planned to be carried out in remote areas with harsh climatic
conditions. Due to the lack of sufficient ground transport infrastructure, cargo delivery to these
regions is currently possible only with the help of air transport. The importance of this study rests
on the lack of adequate methods for the determination of the reference parameters for geodesic
hemispherical dome structures at an early stage of design. In particular, it is common for the issues
regarding the transportation of structural elements as well as those that involve ensuring the strength
and the technological characteristics of the structure to not be considered simultaneously. This study
owes its relevance to the rapid development of the uninhabited territories of the Russian Federation
in the context of the global ecological crisis caused by anthropogenic impact on the environment.
Transportation. Buildings 2022, 12,
242. https://doi.org/10.3390/
buildings12020242
Keywords: geodesic dome; technological design; structural layout; finite element method; stressstrain state; weight analysis
Academic Editor: Pierfrancesco
De Paola
Received: 21 January 2022
Accepted: 16 February 2022
Published: 19 February 2022
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Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
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distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1. Introduction
This article is a continuation of the research cycle devoted to the geodesic dome
structures featured in a previous article that was published in the journal Quality and Life in
December 2021 [1].
The Russian Federation ranks first in the world in terms of its area, which is 17,130,000 km2 .
At the same time, only about 12% of this area is mastered, while there are no traces of
human activity in the rest of the territory. As part of the “Strategy for the long-term
development of the Russian Federation with low greenhouse gas emissions until 2050”
project implementation, by 2030, there is a plan to reach a level of carbon emissions equal
to 67% of 1990 levels, and by 2050, to decrease emissions further to 20% of 1990 levels.
At the same time, there is a plan to build new cities in Siberia, which was officially
announced by the Minister of Defense of the Russian Federation, S.K. Shoigu, on 5 August
2021. According to this statement, three scientific and industrial centers with a population
of three hundred thousand to one million people are planned to be built in Siberia between
Bratsk and Krasnoyarsk, as well as in the area of Kansk and Lesosibirsk.
The creation of settlements in the Russian Federation is also considered within the
framework of the comprehensive scientific and technical program “Connectivity of the
Buildings 2022, 12, 242. https://doi.org/10.3390/buildings12020242
https://www.mdpi.com/journal/buildings
Buildings 2022, 12, 242
2 of 12
territory of the Russian Federation”, selected by the Council for the Priority of Scientific
and Technological Development.
In the context of these two initiatives, the general task of ensuring the ecological preservation of the territories not affected by anthropogenic impact along with their development
is highlighted.
This area has a sharply continental climate, which can be characterized by a long winter
with a temperature drop as low as −50 ◦ C and a short, hot summer with a temperature
high of +30 ◦ C. The hot season in the nearest cities lasts from 240 to 250 days a year, and
in summer, there is a high load on the power grid because of the high energy costs of air
conditioning. Thus, one of the steps necessary to achieve this task is to increase the energy
efficiency of the capital buildings. Salosina, Alifanov, and Nenerokomov in their work [2]
researched high-porosity open-cell foam in terms of its levels of thermal protection.
It is known that dome buildings in such climatic conditions have up to a one and a half
times higher level of energy efficiency when compared to classical rectangular buildings.
In the work of Pogosyan, Strelets, and Vladimirova [3], the communication gap in the
spatial development of the country is highlighted; transport and electronic accessibility is
limited. However, in the work of Lutovinov, Pogosyqn, and Lupyan [4], a solution to the
communication gap is proposed.
The use of dome geodesic structures can provide an alternative solution to this problem, since this can simplify the rapid construction of reliable long-term shelters without the
need for special construction equipment because of the added possibility of transporting
the assembly elements by air to areas that lack a developed transport infrastructure.
At the same time, reducing the mass of such structures will also significantly increase
the maximum range of air transportation. The problems involved in the increasing of the
maximum range of aircraft were previously dealt with by Dolgov and Aruvelli [5].
Additionally, another advantage of these kinds of structures is their high stability
in the event of natural disasters such as earthquakes and hurricanes because of their
geometric shape.
All of these factors taken together allow us to speak about the relevance of a deeper
study of dome structures.
2. Literature Review
A large number of domestic and foreign scientists are currently engaged in research
on the strength and the functional characteristics of dome structures. The current research
was inspired by the work of the following authors:
•
•
•
•
•
•
•
Gritsenko, who was engaged in research on the history of the lattice geodesic dome’s
creation [6].
Maria, Esipova, Vintural, and Shabanov, who, in their works, investigated the possibility of using geodesic domes in various economical fields [7–10].
Granev, Kodysh, Mamin, Bobrov, Reutsu, and Kuznichenko, who studied the existing
lattice structures in terms of their reliability and manufacturability [11].
Gorkoltseva, Demidov, Olshanchenko, Shiryaeva, Romanovich, Shanko, Shishkina,
and Kaloshina in their works [12–16] investigated the advantages of dome structures
in comparison to those of classical buildings, according to the criteria of energy
efficiency, weight, materials cost, air exchange, seismic resistance, wind resistance, and
environmental friendliness.
Zhuravlev, Glushko, and Lakhov in their works [17–20] studied the strength characteristics of geodetic structural layouts.
Lakhova, Gorkoltseva, Miryaeva, and Pilarska in their works [21–24] described various
ways of designing lattice domes.
Chepurnenko conducted a comparative analysis of various designs of wavy domes in
order to identify their advantages and disadvantages [25].
Buildings 2022, 12, x FOR PEER REVIEW
3 of 13
•
•
Buildings 2022, 12, 242
Lakhova, Gorkoltseva, Miryaeva, and Pilarska in their works [21–24] described
Lakhova,
Gorkoltseva,
Miryaeva,
and Pilarska in their works [21–24] described
various
ways
of designing
lattice domes.
various
ways
of
designing
lattice
domes.
3 of 12
•
Chepurnenko conducted a comparative analysis of various designs of wavy domes
• in
Chepurnenko
conducted
a
comparative
analysis
of
various
designs
of
wavy
domes
order to identify their advantages and disadvantages [25].
in order to
identify their
advantages
and disadvantages
[25].
•
Barbieri,
Machado,
Barbieri,
Lima, Rossot,
Guan, Virgin,
and Helm calculated the
• strength
Barbieri, characteristics
Machado, Barbieri,
Lima,
Rossot,
Guan,
Virgin,
and with
Helmthe
calculated
the
of
dome
structures
and
compared
them
results from
•
Barbieri, Machado, Barbieri, Lima, Rossot, Guan, Virgin, and Helm
calculated
the
strength
characteristics
of dome structures and compared them with the results from
the
experimental
data [26,27].
strength
characteristics
of dome structures and compared them with the results from
the
experimental
data
[26,27].
• the
Jihong,
Mingfei, data
Kaveh,
and Talatahari were engaged in the study of dome strucexperimental
[26,27].
Jihong,
Mingfei,
Kaveh,
and
Talatahari
were
engaged
intasks
the study
oftodome
structures
inMingfei,
terms ofKaveh,
their
strength
characteristics,
as well
related
design
opand
Talatahari
were
engaged
in as
the
study
of dome
structures
•• Jihong,
tures
in
terms
of
their
strength
characteristics,
as
well
as
tasks
related
to
design
optimization
[28,29].
in
terms of their
strength characteristics, as well as tasks related to design optimizatimization
[28,29].
tion
Such[28,29].
a wide interest in the subject of the study, as well as the lack of a comparative
Suchofaawide
wide
interestof
ingeodesic
thesubject
subject
ofthe
thestudy,
study,
aswell
wellas
asthe
the
lackofofjustifies
comparative
Such
interest
in
the
of
as
lack
aacomparative
analysis
the
varieties
hemispherical
three-layered
domes,
the relanalysis
of
the
varieties
of
geodesic
hemispherical
three-layered
domes,
justifies
therelerelanalysis
of
the
varieties
of
geodesic
hemispherical
three-layered
domes,
justifies
the
evance of the study. In regions with a mild climate, a single-layer structural layout
(Figevance
of
the
study.
In
regions
with
a
mild
climate,
a
single-layer
structural
layout
(Figvance
study.used.
In regions
climate,
a single-layer
layout (Figure
1)
ure 1)ofisthe
usually
Most with
often,a mild
natural
materials
are used structural
in the structural
elements,
ure
1) is the
usually
used.
Mostnatural
often,
natural
materials
used
in
the structural
isnamely,
usually
used.
Most
often,
are used
in the
structural
elements,
structural
frame
of the materials
dome structure
isare
made
of
wooden
boards. elements,
Innamely,
such a
namely,
structural
frame
of the
structure
made of
wooden
Inlimited
such
the
structural
frame of
the
dome
structure
is
made insulation
ofiswooden
boards.
Inboards.
such
a case,
thea
case,
the the
geometric
characteristics
ofdome
the thermal
material
are
always
case,
the
geometric
characteristics
of
the
thermal
insulation
material
are
always
limited
geometric
characteristics
of the thermal insulation material are always limited by the wood
by the wood
blanks’ parameters.
by the wood
blanks’ parameters.
blanks’
parameters.
Figure 1. Single-layered dome structure without thermal insulation.
Figure1.1.Single-layered
Single-layereddome
domestructure
structurewithout
withoutthermal
thermalinsulation.
insulation.
Figure
As a rule, standard rectangular boards are chosen as the material of structural frame
Asaarule,
rule,
standard
rectangular
boards
arechosen
chosen
asthe
thematerial
materialof
ofstructural
structural
frame
As
standard
rectangular
boards
are
as
frame
elements
(Figure
2), which
have the
following
geometric
characteristics
“L”—length,
elements
(Figure
2),
which
have
the
following
geometric
characteristics
“L”—length,
elements
(Figure
2),
which
have
the
following
geometric
characteristics
“L”—length,
“h”—
“h”—height, and “b”—width.
“h”—height,
and “b”—width.
height,
and “b”—width.
Figure
Figure2.2.Geometric
Geometriccharacteristics
characteristicsof
ofaarectangular
rectangularboard.
board.
Figure 2. Geometric characteristics of a rectangular board.
Since,
achieve
greater
structural
strength,
the boards
are positioned
in such
Since,ininorder
ordertoto
achieve
greater
structural
strength,
the boards
are positioned
in
Since,
in
order
to
achieve
greater
structural
strength,
the
boards
are
positioned
in
asuch
wayathat
narrow
and long
looks
of the
there isthere
a limitation
on
waythe
that
the narrow
andside
long“J”
side
“J” out
looks
out structure,
of the structure,
is a limitasuch
a way that
thethermal
narrow insulation
and long side
“J” looks
of layer
the structure,
is aexceed
limitathe
thickness
of the
material
layer.out
The
thicknessthere
cannot
the width “b” of the boards used because it is necessary to install the inner skin to fix it. At
the same time, the use of very wide boards has the potential to significantly increase the
cost and mass characteristics of the structure, while the use of more affordable materials
with smaller overall dimensions is more effective.
Buildings 2022, 12, 242
tion
tion on
on the
the thickness
thickness of
of the
the thermal
thermal insulation
insulation material
material layer.
layer. The
The layer
layer thickness
thickness cannot
cannot
exceed
exceed the
the width
width “b”
“b” of
of the
the boards
boards used
used because
because itit is
is necessary
necessary to
to install
install the
the inner
inner skin
skin to
to
4 of in12
fix
it.
At
the
same
time,
the
use
of
very
wide
boards
has
the
potential
to
significantly
fix it. At the same time, the use of very wide boards has the potential to significantly increase
crease the
the cost
cost and
and mass
mass characteristics
characteristics of
of the
the structure,
structure, while
while the
the use
use of
of more
more affordable
affordable
materials
materials with
with smaller
smaller overall
overall dimensions
dimensions is
is more
more effective.
effective.
In
a harsh
climate,
where
it isititnecessary
to increase
the distance
between
In
regions
with
aa harsh
climate,
where
is
to
the
beInregions
regionswith
with
harsh
climate,
where
is necessary
necessary
to increase
increase
the distance
distance
bethe
external
and theand
internal
skin of skin
the
structures
are used
forused
laying
tween
the
the
of
dome,
structures
are
for
tween
the external
external
and
the internal
internal
skindome,
of the
themultilayer
dome, multilayer
multilayer
structures
are
used
for
thermal
insulation
material,
in
which
sectional
structures
are
used
as
structural
elements
laying
thermal
insulation
material,
in
which
sectional
structures
are
used
as
structural
laying thermal insulation material, in which sectional structures are used as structural
instead
of instead
conventional
boards (Figure
3).(Figure
elements
boards
elements
instead of
of conventional
conventional
boards
(Figure 3).
3).
Figure
Geometrical
characteristics
of
the
sectional
structural
element.
Figure3.3.
3.Geometrical
Geometricalcharacteristics
characteristicsof
ofthe
thesectional
sectionalstructural
structuralelement.
element.
Figure
In
this
case,
the
elements
of
the
inner
and
outer
layers
of
the
dome
“X”
are
connected
Inthis
thiscase,
case,the
theelements
elementsof
ofthe
theinner
innerand
andouter
outerlayers
layersof
ofthe
thedome
dome“X”
“X”are
areconnected
connected
In
using
the
structural
elements
in
“Y”,
which
keep
the
layers
at
a
certain
distance,
thereby
using
the
structural
elements
in
“Y”,
which
keep
the
layers
at
a
certain
distance,
thereby
using the structural elements in “Y”, which keep the layers at a certain distance, thereby
providing
the
necessary
thickness
for
holding
a
thicker
layer
of
thermal
insulation
mateprovidingthe
thenecessary
necessary
thickness
holding
a thicker
layer
of thermal
insulation
mateproviding
thickness
forfor
holding
a thicker
layer
of thermal
insulation
material.
rial.
An
example
of
such
a
structural
layout
is
shown
in
Figure
4.
rial.
An example
of asuch
a structural
is shown
in Figure
4.
An
example
of such
structural
layoutlayout
is shown
in Figure
4.
Figure
Two-layer
geodesic
dome
structure.
Figure4.4.
4.Two-layer
Two-layergeodesic
geodesicdome
domestructure.
structure.
Figure
The
disadvantage
of
using
such
modular
structural
elements
is
their
low
rigidity,
Thedisadvantage
disadvantageof
ofusing
usingsuch
suchmodular
modular structural
structural elements
elements is
is their
their low
low rigidity,
rigidity,
The
owing
to
the
ItIt is
to
owingto
tothe
the fact
fact that
that they
set
of
rectangles
that
have
four
sides.
is possible
possible
to ininowing
they are
are aa set
setof
ofrectangles
rectanglesthat
thathave
havefour
foursides.
sides.
It
is
possible
to
crease
using
structural
layout
consisting
of
elements.
There
increase
therigidity
layout
consisting
of triangular
elements.
There
are
crease the
the
rigidity by
byusing
usingaaastructural
structural
layout
consisting
of triangular
triangular
elements.
There
are
two
of
structures
structural
that
of
two
of geodesic
domedome
structures
structural
layoutslayouts
that consist
only ofonly
triangular
are variants
two variants
variants
of geodesic
geodesic
dome
structures
structural
layouts
that consist
consist
only
of tritriangular
elements.
They
are
known
and
used
in
many
countries.
elements.
They
are
known
and
used
in
many
countries.
angular elements. They are known and used in many countries.
3. Materials and Methods
To determine the advantages and the disadvantages of two-layer hemispherical
geodesic dome structural layouts with a triangular structure, a series of strength calculations for different variants using the finite element method was carried out.
The finite element method makes it possible to analyze the stress-strain state of
structures with a high degree of accuracy. Aabid, Zakuan, Khan, and Ibrahim [30] claimed
that this method allows the analysis of the structures of different scales, which makes this
method universal.
Buildings 2022, 12, 242
3. Materials
and Methods
To determine
the advantages and the disadvantages of two-layer hemispherical
To
determine
the advantages
and
the disadvantages
two-layer
hemispherical
geodesic dome structural
layouts with
a triangular
structure, of
a series
of strength
calculageodesic
dome
structural
layouts
with
a
triangular
structure,
a
series
of
strength
calculations for different variants using the finite element method was carried out.
tionsThe
for different
variants
using the
finite
was the
carried
out.
finite element
method
makes
it element
possiblemethod
to analyze
stress-strain
state of
The finite
methodofmakes
it possible
analyze Khan,
the stress-strain
state
of
structures
with element
a high degree
accuracy.
Aabid,toZakuan,
and Ibrahim
[30]
5 of 12
structures
high degree
Aabid,
Zakuan, of
Khan,
and scales,
Ibrahim
[30]
claimed
thatwith
this amethod
allows of
theaccuracy.
analysis of
the structures
different
which
claimed
that
this method
allows the analysis of the structures of different scales, which
makes
this
method
universal.
makes
this
method
universal.
The strength and weight parameters were jointly considered, depending on the
The
andand
weight
parameters
were jointlyjointly
considered,
depending
on the amount
Thestrength
parameters
considered,
depending
on the
amount
ofstrength
change in
theweight
thickness
of the bar were
filler and the number
of structural
elements.
ofamount
changeof
in change
the thickness
of
the
bar
filler
and
the
number
of
structural
elements.
the bar
filler andlayout
the number
of structuralThe
elements.
Four variants in
of the
the thickness
geodesic of
domes’
structural
were considered.
strucFour variants
of of
thethe
geodesic
domes’
structural
layout
werewere
considered.
The structures
geodesic
domes’
structural
layout
considered.
The
structures Four
undervariants
consideration
were spatial
three-dimensional
three-layer
structures
consistunder
consideration
were spatial
three-dimensional
three-layer
structures
consisting
of
tures
under
consideration
were
spatial
three-dimensional
three-layer
structures
consisting of load-bearing layers in the form of polygonal grids and a bar filler. The structural
load-bearing
layers
in
the
form
of
polygonal
grids
and
a
bar
filler.
The
structural
layouts
ing of load-bearing
layers in were
the form
of polygonal
a barwere
filler.
The structural
layouts
under consideration
divided
into two grids
types,and
which
further
divided
under
consideration
were
divided
into
two
types,
which
were
further
divided
into
two
layouts
consideration
were both
divided
into layers
two types,
which were
further
divided
into
two under
variants.
In the first type,
bearing
are polygonal
grids
consisting
of
variants. In the first type, both bearing layers are polygonal grids consisting of triangles
into two (Figure
variants.5),Inand
the in
first
bothtype,
bearing
are polygonal
consisting
of
triangles
thetype,
second
the layers
first layer
consists ofgrids
triangles
and the
(Figure 5), and in the second type, the first layer consists of triangles and the second of
triangles
5), (Figure
and in the
second
type,
first layer
consistswere
of triangles
and the
second
of (Figure
hexagons
6). In
addition
to the
hexagons,
pentagons
also present
in
hexagons (Figure 6). In addition to hexagons, pentagons were also present in the model,
second
of hexagons
6).was
In addition
tofor
hexagons,
pentagons
werestructure
also present
in
the
model,
since this (Figure
condition
necessary
the closure
of the dome
at the
since this condition was necessary for the closure of the dome structure at the pole.
the
model,
since
this
condition
was
necessary
for
the
closure
of
the
dome
structure
at
the
pole.
pole.
Figure 5. First variant of the structural layout.
Figure 5. First variant of the structural layout.
Figure 5. First variant of the structural layout.
Figure 6. Second variant of the structural layout.
Figure6.6.Second
Secondvariant
variantofofthe
thestructural
structurallayout.
layout.
Figure
In the first type of structure, two variants differ regarding the orientation of the fillthe
first
typeofofstructure,
structure,
two
differ
regarding
the
orientation
the
fillInIn
the
first
type
two
variants
differ
regarding
thefiller
orientation
of of
theonly
filler.
er. In
the
first
variant
the first type
ofvariants
structure
(Figure
7), the
is connected
to
er.
In
the first
variant
ofouter
the
type
of
structure
7),
filler
onlyto
to
In
the
first
variant
the
firstfirst
type
of structure
thethe
filler
is is
connected
the
nodal
points
ofofthe
bearing
layer,
and(Figure
in(Figure
the 7),
second
variant
ofconnected
the firstonly
type
of
the
nodal
points
of
the
outer
bearing
layer,
and
in
the
second
variant
of
the
first
type
of
the
nodal
points
of
the
outer
bearing
layer,
and
in
the
second
variant
of
the
first
type
of
structure (Figure 8), the filler is connected to the nodal points of the inner bearing layer.
structure
(Figure
filler
is connected
to the
nodal
points
of the
inner
bearing
layer.
structure
(Figure
8),8),
the
filler
is connected
the
nodal
points
of the
inner
bearing
layer.
In
In
the first
variant
ofthe
the
second
type oftostructure(
Figure
9),
the
layer
consisting
of
triIn first
the is
first
variant
ofsecond
the second
type
of hexagons
structure(isFigure
theconsisting
layer
of trithe
variant
of the
type of
structure(Figure
9),
the 9),
layer
of triangles
angles
external,
the
layer
consisting
of
internal,
and
in theconsisting
second
variant
isof
external,
thetype
layerof
consisting
of hexagons
is internal,
and in the
of the
angles
is external,
the
layer consisting
of hexagons
is internal,
andsecond
in the variant
second variant
the second
structure
(Figure
10),
vice
versa.
second
type
of some
structure
(Figureof10),
vicestructures,
versa.
of the
second
type of
structure
(Figure
10),
vice versa.
Here
are
examples
dome
parts of which were transported by air:
Here
are
some
examples
ofofdome
structures,
parts
ofofwhich
were
transported
byby
air:
Here
are
some
examples
dome
structures,
which
were
transported
air:
•
The metal geodesic dome of the US Antarcticparts
Station
Amundsen–Scott
is located
at
• • The
metal
geodesic
dome
of
the
US
Antarctic
Station
Amundsen–Scott
is
located
at
The
metal
geodesic
dome
of
the
US
Antarctic
Station
Amundsen–Scott
is
located
at
the South Pole of the Earth. It was built in 1975 and used to be the main shelter of the
the
South
Pole
of
the
Earth.
It
was
built
in
1975
and
used
to
be
the
main
shelter
of
the
the
South
Pole
of
the
Earth.
It
was
built
in
1975
and
used
to
be
the
main
shelter
of
the
research station. Its service life in the extreme climatic conditions of the South Pole
research
station.
ItsIts
service
lifelife
in the
extreme
climatic
conditions
of theofSouth
Pole was
research
station.
service
in the
extreme
climatic
conditions
the South
Pole
more than 30 years. This dome structure was built from elements premanufactured in
the factory and delivered by air transport.
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was more than 30 years. This dome structure was built from elements premanufac6 of 13
tured in the factory and delivered by air transport.
6 of 12
was more than 30 years. This dome structure was built from elements premanufactured in the factory and delivered by air transport.
Figure 7. Metal geodesic dome of the US Antarctic Station Amundsen–Scott.
The geodesic dome of the DYE-2 station of the missile strike early detection system
in Greenland, with a diameter of more than 20 m, the construction of which began in the
late 1950s. The construction of this structure was carried out from parts transported exFigure 7. Metal geodesic dome of the US Antarctic Station Amundsen–Scott.
clusively by air transport.
The geodesic dome of the DYE-2 station of the missile strike early detection system
in Greenland, with a diameter of more than 20 m, the construction of which began in the
late 1950s. The construction of this structure was carried out from parts transported exFigure 7.
7. Metal
Metal geodesic
geodesic dome
dome of
of the
the US
US Antarctic
Antarctic Station
Station Amundsen–Scott.
Amundsen–Scott.
Figure
clusively
by air transport.
The geodesic dome of the DYE-2 station of the missile strike early detection system
in Greenland, with a diameter of more than 20 m, the construction of which began in the
late 1950s. The construction of this structure was carried out from parts transported exclusively by air transport.
Figure 8. Geodesic dome of the DYE-2 station of the missile strike early detection system in
Greenland.
•
As a close analogue to the first type of structure, we may mention the lighting dome
of the Research Institute of Building Physics (NIISF), which was built in Moscow in
Figure 8.8.Geodesic
Geodesic
dome
of DYE-2
the DYE-2
station
of the strike
missile
strike
early system
detection
system in
Figure
dome
of the
station
of the missile
early
detection
in Greenland.
1981.
Greenland.
•
As a close analogue to the first type of structure, we may mention the lighting dome
of the Research Institute of Building Physics (NIISF), which was built in Moscow in
Figure 8. Geodesic dome of the DYE-2 station of the missile strike early detection system in
1981.
Greenland.
•
As a close analogue to the first type of structure, we may mention the lighting dome
of the Research Institute of Building Physics (NIISF), which was built in Moscow in
1981.
Figure 9.
9. The
The Lighting
Lighting dome
dome of
of the
the NIISF,
NIISF,outside
outsideview.
view.
Figure
The geodesic dome of the DYE-2 station of the missile strike early detection system in
Greenland, with a diameter of more than 20 m, the construction of which began in the late
1950s. The construction of this structure was carried out from parts transported exclusively
Figure
9. The Lighting dome of the NIISF, outside view.
by
air transport.
•
As a close analogue to the first type of structure, we may mention the lighting dome of
the Research Institute of Building Physics (NIISF), which was built in Moscow in 1981.
•
As a close analogue to the second type of structure, we may mention the geodesic
dome of the museum dedicated to the environment and water resources in Montreal (Canada), built of steel rods in 1967, with a height of 62 m and a diameter
of 76 m, designed by American engineer and architect Richard Buckminster Fuller
(Figures 11 and 12).
Figure 9. The Lighting dome of the NIISF, outside view.
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77 of
13
Figure 10. The Lighting dome of the NIISF, inside view.
•
As a close analogue to the second type of structure, we may mention the geodesic
dome of the museum dedicated to the environment and water resources in Montreal
(Canada), built of steel rods in 1967, with a height of 62 m and a diameter of 76 m,
designed by American engineer and architect Richard Buckminster Fuller (Figures
Figure
10.
The12).
Lighting dome
dome of
of the
the NIISF,
NIISF, inside
inside view.
view.
and
Figure11
10.
The
Lighting
•
As a close analogue to the second type of structure, we may mention the geodesic
dome of the museum dedicated to the environment and water resources in Montreal
(Canada), built of steel rods in 1967, with a height of 62 m and a diameter of 76 m,
designed by American engineer and architect Richard Buckminster Fuller (Figures
11 and 12).
Figure11.
11. Montreal
Montreal“biosphere,”
“biosphere,”general
generalview.
view.
Figure
Figure 11. Montreal “biosphere,” general view.
Figure12.
12. Montreal
Montreal“biosphere”
“biosphere”structure.
structure.
Figure
The general
general views
views of
of the
the geodetic
geodetic hemispherical
hemispherical structural
structural layouts
layouts considered
considered are
are
The
presentedin
inFigures
Figures13–16.
13–16.
presented
For each of the four design versions, ten models were created with the division of the
main triangle edge into a different number of structural elements from 2 to 20 in increments
Figure
Montreal
“biosphere”
structure.
of
two.12.
Each
structural
element
is a circular beam with a radius of 20 mm.
Next, a strength analysis of all forty models was carried out with a change in the
Thebetween
general the
views
of the
geodetic
layouts of
considered
are
distance
bearing
layers
from hemispherical
200 mm to 500 structural
mm in increments
50 mm. As
a
presented
in
Figures
13
–
16.
result, 280 calculations were carried out.
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8 of 13
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Figure13.
13.Type
Type1,1,variant
variant1.1.
Figure
Figure 13. Type 1, variant 1.
Figure 13. Type 1, variant 1.
Figure 14. Type 1, variant 2.
Figure 14. Type 1, variant 2.
Figure 14. Type 1, variant 2.
Figure 14. Type 1, variant 2.
Figure 15. Type 2, variant 1.
Figure 15. Type 2, variant 1.
Figure 15. Type 2, variant 1.
Figure 15. Type 2, variant 1.
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Figure16.
16.Type
Type2,2,variant
variant2.2.
Figure
For eachcalculations
of the fourwere
design
versions,
ten models
were
created
with the
division
Strength
carried
out using
the finite
element
method
in the
Hyper-of
the main
triangle
edgeininto
a different
of structural
from that
2 todiffered
20 in inMesh
software
package
a static
solver. number
Four groups
of modelselements
were created
of two.
Eachper
structural
is a triangle
circularfrom
beam2 with
radius
of 20
mm.
increments
the number
of edges
the sideelement
of the main
to 20.aThe
model
parameters
Next,
a
strength
analysis
of
all
forty
models
was
carried
out
with
a
change
in the
are given in Table 1.
distance between the bearing layers from 200 mm to 500 mm in increments of 50 mm. As
a result,
280 calculations
carried out.
Table
1. Parameters
for FEM were
models.
Strength calculations were carried out using the finite element method in the HyParameter
perMesh software
package in a static solver. Four groups of Value
models were created that
Dome
diameter,
m per the side of the main triangle 10
differed in the
number
of edges
from 2 to 20. The model
Section view
the structural
elements
Circle
parameters
are of
given
in Table 1.
Structural elements section radius, mm
As mentioned previously, 280 calculations were carried out20for each of the models
Material
Steel
that was created
manually.
The use of the highly convenient
HyperMesh software
Young’s Modulus,
MPa
2,100,000
package made it
possible
to
significantly
reduce
the
preparation
time
Poisson ratio
0.3 of the model.
3
Depending
on
the
geometric
division
of
the
main
triangle,
the
number of structural
7700
Density, kg/m
Element
type
B31—linear
spatial
beam
finite
elementinto 10
elements (beams) changed, but regardless of the division, each beam was
divided
finite elements. Thus, the structural elements had the same mesh density. This partition
wasAs
dictated
primarily
by the requirement
of mesh
mentioned
previously,
280 calculations
wereconvergence.
carried out for each of the models
The
final
elements
of
the
Bar-type,
instead
of the Rod-type,
were used
in the
model;
that was created manually. The use of the highly convenient
HyperMesh
software
package
the
elements
of
the
Bar-type
are
able
to
work
not
only
in
tension-compression,
but
also in
made it possible to significantly reduce the preparation time of the model.
bending.
Depending on the geometric division of the main triangle, the number of structural
The(beams)
structurechanged,
under itsbut
own
weight was
calculated.
For
this,beam
the gravity
acceleration
elements
regardless
of the
division,
each
was divided
into
2. As boundary conditions, a restriction on all degrees of freedom for the
was
set
to
9.8
m/s
10 finite elements. Thus, the structural elements had the same mesh density. This partition
lower
row ofprimarily
nodes was
simulating of
themesh
presence
of a foundation.
was
dictated
by used,
the requirement
convergence.
The final elements of the Bar-type, instead of the Rod-type, were used in the model;
Table
1. Parameters
FEM models.
the
elements
of the for
Bar-type
are able to work not only in tension-compression, but also
in bending.
Parameter
Value
The structure under its own weight was calculated. For this, the gravity acceleration
Dome diameter, m
10
was set to 9.8 m/s2 . As boundary conditions, a restriction on all degrees of freedom for the
Section view of the structural elements
Circle
lower row of nodes was used, simulating the presence of a foundation.
Structural elements section radius, mm
20
Material
Steel
4. Results and Discussion
Young’s
Modulus,
MPa
2100,000
The stress-strain state of the structures was estimated by the values of stresses and
ratiofor all models do not exceed the yield strength
0.3
displacements. The Poisson
stress values
of structural
3
Density,ofkg/m
7700 17 shows the
steels. The peak values
the displacements were systemized. Figure
type values on the massB31—linear
spatial
finite
element
dependences of the Element
displacement
of the structure
andbeam
the wall
thickness.
4. Results and Discussion
The stress-strain state of the structures was estimated by the values of stresses and
displacements. The stress values for all models do not exceed the yield strength of
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Figure 17. Summary graph of the peak displacements and the mass of the structure on the reference
Figure 17. Summary graph of the peak displacements and the mass of the structure on the reference
parameters of the structural layout for geodesic hemispherical two-layer dome structures with a
parameters
of the structural layout for geodesic hemispherical two-layer dome structures with a
core filler.
core filler.
5. Conclusions
The parameters accepted in the finite element model correspond to the parameters
Based onthat
thecan
dependencies
during
the analysis
of the and
different
variants of of structural
be used in aobtained
real design.
Changing
the material
the cross-section
geodesic hemispherical
dome
structures,
an
algorithm
was
determined
for
selecting
a
elements would yield different results, but this is the topic of another separate
study.
rational combination
of
strength
and
weight
characteristics
taking
into
consideration
the
In this paper, a comparative analysis of the structural layout of domes was carried out.
limitations associated
with manufacturability
and the
of structural
materials.
After analyzing
the graphs in Figure
17,choice
a rational
design can
be chosen based on the
The algorithm
developed
for
the
selection
of
reference
parameters
during
the initial
requirements set.
stage of the designIfof
geodetic
hemispherical
dome
structures
may
also
facilitate
task to choose a
the basic requirement is air transportation possibility, then it is the
advisable
of transporting
the
elements
of
such
structures
by
air
in
the
future.
design that meets all other restrictions with a minimum weight. Judging by the graphs, the
The desired
outcome
is atotype
ensure
the possibility
of building
reliable,
rational
choiceofinthis
thisstudy
case is
2 structure
with a division
of six
elements per main
long-term shelters
in asince
short
withthe
thedisplacements
help of only small
of have
workers,
and low values.
triangle,
in time,
this case,
of the groups
structure
extremely
without the use
special
construction
by based
ensuring
a rational
combination
of
Theofwall
thickness
should equipment,
be determined
on the
required
type of insulation
material.
the structuralSo,
and
technological
parameters
ofminimum
the structures
is used
with
minimum
when
using mineral
cotton, the
thickness
would
be a350
mm; at the same time,
mass of material.
when using polyurethane, it is possible to limit the wall thickness to 200 mm.
Similarly, a design selection can be made that is guided by other requirements.
Author Contributions: Conceptualization, A.K.; methodology, O.D.; validation, S.P.; formal analysis, V.K.; investigation, A.K.; writing—original draft preparation, S.P.; writing—review and edit-
Buildings 2022, 12, 242
11 of 12
Two types of structures have been worked out in two variants. Structures of the second
type have better mass-stiffness characteristics, while structures of the first type are better
in terms of their manufacturability. In particular, structures of this type require less labor
intensity during assembly because of the presence of the horizontal rows of the structural
elements. As a result of our calculations, it became possible to quantify the gain in weight
of the second type of structure relative to the first one, which allows the designer to make
an optimal choice in favor of a design of one type over another.
5. Conclusions
Based on the dependencies obtained during the analysis of the different variants of
geodesic hemispherical dome structures, an algorithm was determined for selecting a
rational combination of strength and weight characteristics taking into consideration the
limitations associated with manufacturability and the choice of structural materials.
The algorithm developed for the selection of reference parameters during the initial
stage of the design of geodetic hemispherical dome structures may also facilitate the task of
transporting the elements of such structures by air in the future.
The desired outcome of this study is to ensure the possibility of building reliable,
long-term shelters in a short time, with the help of only small groups of workers, and
without the use of special construction equipment, by ensuring a rational combination of
the structural and technological parameters of the structures is used with a minimum mass
of material.
Author Contributions: Conceptualization, A.K.; methodology, O.D.; validation, S.P.; formal analysis,
V.K.; investigation, A.K.; writing—original draft preparation, S.P.; writing—review and editing, V.Z.;
visualization, V.A.; supervision, O.D. All authors have read and agreed to the published version of
the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: The data presented in this study are available on request from the
corresponding author.
Conflicts of Interest: The authors declare no conflict of interest.
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